Posts Tagged 'Logic'

Berry’s Paradox and Gödel’s Incompleteness Theorem

 

Chaitin-Boolos

Left: Argentine-American mathematician
Gregory Chaitin [image from here]. Right: American philosopher and logician
George Boolos [image Wikimedia Commons].

A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form

THE SMALLEST NATURAL NUMBER THAT CANNOT BE
DEFINED IN FEWER THAN TWENTY WORDS.

Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’

Buridan’s Ass

Jean-Buridan

Jean Buridan (c. 1300-1360).

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

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The Ross-Littlewood Paradox

Ross-Littlewood-01

Ross-Littlewood Paradox [Image from Steemit website:  here. ]

A most perplexing paradox appeared in Littlewood’s book A Mathematician’s Miscellany. It was later analysed in detail by Sheldon Ross in his 1988 book A First Course in Probability.

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The “Napoleon of Crime” and The Laws of Thought

NewLightOnGBooleA fascinating parallel between a brilliant mathematician and an arch-villain of crime fiction is drawn in a forthcoming book – New Light on George Boole – by Des MacHale and Yvonne Cohen. Professor James Moriarty, master criminal and nemesis of Sherlock Holmes, was described by the detective as “the Napoleon of crime”. The book presents convincing evidence that Moriarty was inspired by Professor George Boole [TM151, or search for “thatsmaths” at irishtimes.com].

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“Dividends and Divisors Ever Diminishing”

Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week’s ThatsMaths post]

Joyce-in-Zurich

Joyce in Zurich: did he meet Zermelo?

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Enigmas of Infinity

Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com]. 

infinity-symbols

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The Shaky Foundations of Mathematics

The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].

platoaristotlepythagoraseuclid

Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]

The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.

Continue reading ‘The Shaky Foundations of Mathematics’

Peano Music

The links between mathematics and music are manifold. Mathematics can be set to music in a simple but surprising manner. For the award ceremony of the Gödel Medal in 2014, a musical interpretation of Gödel’s incompleteness Theorems was written by Danish composer Niels Marthinsen. It encodes the basic axioms of number theory that form the focus of Gödel’s Theorems.

Peano-Axioms

The Peano Axioms in symbolic form.

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The Year of George Boole

This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen’s College Cork.

Boole-Year-UCC-Small Continue reading ‘The Year of George Boole’

Degrees of Infinity

Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing.

Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).

Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).

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Invention or Discovery?

Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:

        Conjecture: Definitions are invented. Theorems are discovered.

The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’

The Loaves and the Fishes

 One of the most amazing and counter-intuitive results in mathematics was proved in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. Banach was a mathematical prodigy, and was the founder of modern functional analysis. Tarski was a logician, educated at the University of Warsaw who, according to his biographer, “changed the face of logic in the twentieth century” through his work on model theory.

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